In my PDE course we discuss the Hamilton- Jacobi equation and have briefly mentioned viscosity solutions. My question concerns a specifiy example in dimension $d =1$:
$\partial_t u(x,t) + \frac{1}{2}(\partial_x u(t,x))^2 = 0$ with initial condition $u(0,x) = |x|$ on $x \in [-1,1]$ and $u$ $2$-periodic on $\mathbb{R}\setminus [-1,1]$, so comprising in total a zig-zag function over the real numbers.
Goal: we try to prove that $u(t,x) = u(0,x) - \frac{t}{2}$ is not a viscosity solution. $u(t,x)$ is a solution on $(t,x) \in \mathbb{R}\times \mathbb{R}\setminus \mathbb{Z}.$ A hint is given that problems should arise at the minimums of $u(0,x),$ so at $2k+1, k \in \mathbb{Z.}$
My questions:
- Why do we know that problems would arise there?
- How do we find a $\phi \in C^{\infty}(\mathbb{R_{\geq}\times \mathbb{R}})$ that proves that $u(t,x)$ is not a supersolution?
So far, I've looked up Chapter 10 in Evans' book on PDEs and questions like this one: Desperate on Viscosity (Sub)solutions, but I'm clueless.
